A nonlinear stability analysis of aircraft controller characteristics 2014-08-07¢ A...
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A nonlinear stability analysis of aircraft controller characteristics
outside the standard flight envelope
Stephen J. Gill, Mark H. Lowenberg and Simon A. Neild Faculty of Engineering, University of Bristol, Bristol, BS8 1TR, UK
Luis G. Crespo National Institute of Aerospace, 100 Exploration Way,
Hampton VA 23666 USA
Bernd Krauskopf Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
Guilhem Puyou Airbus France, 31060 Toulouse Cedex 03, France
In this paper we evaluate the influence of the flight control system over the off-nominal flying dynamics of a remotely operated air vehicle. Of particular interest is the departure/upset behaviour of the closed- loop system near stall. The study vehicle is the NASA Generic Transport Model and both fixed-gain and gain-scheduled versions of a LQR-PI controller are evaluated. The differences in effectiveness of the controllers are first demonstrated with time responses to fixed commands. Then, bifurcation analysis is used to characterize spiral and spin behaviour of the aircraft in both open- and closed-loop configurations. This yields an understanding of the underlying vehicle dynamics outside the standard flight envelope, hence it provides a means of assessing the effectiveness of the controller and evaluating the upset tendencies of the aircraft.
The use of feedback control systems in civil aircraft is well established, with relatively complex control laws in place on modern fly-by-wire airliners; see  for example. These are designed for nominal flight conditions where the aircraft is close to symmetric trimmed or manoeuvring flight, and incorporate envelope-limiting functions to attempt to prevent excursions beyond the nominal flight envelope. The control law design process typically utilises linear methods, often based on linear time-invariant (LTI) models, even if the final controller is nonlinear by virtue of scheduling between gains designed at specific operating points.
Despite such control laws, that aim to maintain a safe operating envelope, the problem of airliners en- countering upset phenomena leading to loss-of-control (LOC) does exist . This is characterised by a flight trajectory involving large deviations in attitude and angular rates. During such excursions, the aircraft dynam- ics are dominated by aerodynamic forces and torques that are strongly nonlinear and might be coupled with the moments of inertia and pilot commands. It is often of interest to determine if the flight controller plays a beneficial role on the recovery from vehicle upsets. Generally, LTI models are, however, of limited value when analysing nonlinear systems. Unfortunately, most of the mathematical tools for control analysis are based on such models .
A variety of control design approaches have been developed to overcome the limitations of linear controllers. A popular approach is gain-scheduling, where a set of linear controllers are designed at several operating points and then blended together with respect to an input or state variable . They do, however, usually suffer the limitation of being designed only at trim points within the expected operating envelope and, hence, may not cope well with nonlinearity under off-nominal conditions. Also, care needs to be exercised when scheduling with respect to a state variable that changes rapidly. To overcome the limitations of linearity, much work has been done on nonlinear control law design, such as by dynamic inversion  and a host of adaptive techniques [6, 7]. These are also not necessarily problem-free, with issues such as robustness to model uncertainty and stability guarantees requiring special attention. Typically, nonlinear controllers are also evaluated only under nominal operating conditions (i.e. those within the expected operating range, and up to the stall), partly because this is how the design requirements are stipulated.
Whether implementing gain-scheduled control or another nonlinear approach, a suitably representative math- ematical model is needed. Due to the increased nonlinearity and time-dependence inherent in flight dynamics behaviour under off-nominal conditions – particularly at high angles of attack and/or sideslip – such models are more difficult and expensive to create than conventional models and will inevitably incorporate far higher levels of uncertainty. Hence, they do not generally exist for civil aircraft. However, researchers at NASA Langley have created the Generic Transport Model (GTM): this represents a dynamically scaled version of a twin-engine jet vehicle with a geometry characteristic of low-wing airliners with under-wing-mounted engines . The purpose of the model, derived from a comprehensive wind tunnel test campaign as well as from remotely piloted tests of a powered flight vehicle, was indeed to generate a representative wide-envelope airliner flight dynamics model for the simulation and analysis of upset and LOC . A similar model was created for the EU ‘Simulation of UPset Recovery in Aviation’ (SUPRA) research programme .
To gain insight into the behaviour of a nonlinear dynamical model, it is necessary to understand its ‘struc- ture’: its multiple steady state solution branches, their stability and changes thereof (at bifurcation points) and the dependence of these structures on parameter variations. Whilst the generation of time history simulations from such models is useful, it provides very limited information on the above. Numerical bifurcation analysis, on the other hand, implemented via continuation algorithms , has been shown to be extremely useful in facilitating an understanding of solution characteristics in state-parameter space, allowing response types to be inferred not only locally but also ‘globally’ across a wide operating region .
One field in which continuation and bifurcation analysis has indeed demonstrated significant value is that of nonlinear flight dynamics of aircraft. Since the earliest applications, for example in Carroll and Mehra , Guicheteau  and Zagaynov and Goman , it has been applied to a number of military aircraft models, showing the development of phenomena such as wing rock, inertial coupling and spins (e.g. ) and the influence of highly augmented control systems (e.g. ). Ref.  provides a comprehensive survey of bifurcation analysis applied to fixed-wing flight dynamics problems. This approach has recently been implemented on the NASA Generic Transport Model (GTM)  and the SUPRAmodel  in open-loop form, and has revealed spiral dives and steep spins. An early extension of the approach to closed-loop controllers deployed bifurcation analysis to assess the stability of a wing rock controller with different gains . Bifurcation analysis of closed-loop aircraft has primarily been used to analyse control laws; then the bifurcation diagrams were used to help design control schemes to improve the aircraft behaviour. Littleboy and Smith  used bifurcation analysis to analyse open- loop and then closed-loop versions of the ‘Hypothetical High-Incidence Research Model’ to contribute to control law design through nonlinear dynamic inversion.
Jones et al.  developed the idea of dynamic gain-scheduling where the controller gains were designed as parameters varied within a continuation routine. Here, for each continuation step the trim point is calculated and gains designed by means of a linear quadratic regulator (LQR). This defines a gain schedule with a superior resolution, which is then transformed such that, when scheduled with a fast-varying state, the intended local stability remains valid. However, this method only concerns the aircraft in trim settings and does not assure that off-nominal flight conditions have been stabilised. Bifurcation analysis of the NASA GTM aircraft coupled to a linear parameter varying (LPV) control system was performed by Kwatny et al. . This used bifurcation analysis to perform a rigorous examination of control and regulation around the stall point and showed how controllability is lost at bifurcation points; it also defined safe sets for the aircraft operation and considered recovery from upset in the presence of damage. However, whilst the dynamics of the closed-loop GTM around the stall point was within the scope of this study, controller effects in other off-nominal conditions were not.
A note in  mentions how continuation methods can be used to help tuning the control law. This
idea is extended by Goman and Khramtsovsky  who also analysed a controller designed to suppress wing rock. Here, a two-parameter bifurcation diagram was used to show the combinations of gains for which the controller eliminated wing rock. However, these wing rock controllers use only simple proportional feedback which, although successful for the suppression of wing rock, are not fully representative of controllers typically used on civil transport aircraft. Richardson et al.  extended this idea of continuing controller gains by use of a scalar ‘gain parameter’ which is applied to the state-feedback gains in a feedback-plus-feedforward gain-scheduled controller designed to meet handling qualities objectives by using eigenstructure assignment: it effectively varies a selection of gains simultaneously between zero (when the gain parameter is zero) and their design values (gain parameter unity) and beyond (gain parameter > 1). This study showed that, whilst desired handling qualities were achieved under nominal operating conditions, an undesirable isola that adversely affects system response was identified; it disappeared only when the gain param